New results on approximate Hilbert pairs of wavelet filters with common factors
Sophie Achard, Irène Gannaz, Marianne Clausel, François Roueff
NNew results on approximate Hilbert pairs of wavelet filters withcommon factors
Sophie Achard a , Marianne Clausel b , Ir`ene Gannaz c , and Fran¸cois Roueff da Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France b Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France c Universit´e de Lyon, CNRS UMR 5208, INSA de Lyon, Institut Camille Jordan, France d LTCI, T´el´ecom-Paristech, Universit´e Paris-Saclay, FranceOctober 26, 2017
Abstract
In this paper, we consider the design of wavelet filters based on the
Thiran common-factor approach proposed in Selesnick [2001]. This approach aims at building finite impulseresponse filters of a
Hilbert-pair of wavelets serving as real and imaginary part of a complexwavelet. Unfortunately it is not possible to construct wavelets which are both finitelysupported and analytic. The wavelet filters constructed using the common-factor approachare then approximately analytic. Thus, it is of interest to control their analyticity. Thepurpose of this paper is to first provide precise and explicit expressions as well as easilyexploitable bounds for quantifying the analytic approximation of this complex wavelet.Then, we prove the existence of such filters enjoying the classical perfect reconstructionconditions, with arbitrarily many vanishing moments.
Keywords.
Complex wavelet, Hilbert-pair, orthonormal filter banks, common-factor wavelets
Wavelet transforms provide efficient representations for a wide class of signals. In particularsignals with singularities may have a sparser representation compared to the representation inFourier basis. Yet, an advantage of Fourier transform is its analyticity, which enables to exploitboth the magnitude and the phase in signal analysis. In order to combine both advantagesof Fourier and real wavelet transform, one possibility is to use a complex wavelet transform.The analyticity can be obtained by choosing properly the wavelet filters. This may offer atrue enhancement of real wavelet transform for example in singularity extraction purposes.We refer to Selesnick et al. [2005], Tay [2007] and references therein for an overview of themotivations for analytic wavelet transforms. A wide range of applications can be addressedusing such wavelets as image analysis [Chaux et al., 2006], signal processing [Wang et al.,2010], molecular biology [Murugesan et al., 2015], neuroscience [Whitcher et al., 2005].1 a r X i v : . [ s t a t . M E ] O c t everal approaches have been proposed to design a pair of wavelet filters where one wavelet is(approximately) the Hilbert transform of the other. Using this pair as real and imaginary partof a complex wavelet allows the design of (approximately) analytic wavelets. The simplestcomplex analytic wavelets are the generalized Morse wavelets, which are used in continuouswavelet transforms in Lilly and Olhede [2010]. The approximately analytic Morlet waveletscan also be used for the same purpose, see Selesnick et al. [2005]. However, for practicalor theoretical reasons, it is interesting to use discrete wavelet transforms with finite filters,in which case it is not possible to design a perfectly analytic wavelets. In addition to thefinite support property, one often requires the wavelet to enjoy sufficiently many vanishingmoments, perfect reconstruction, and smoothness properties. Among others linear-phasebiorthogonal filters were proposed in Kingsbury [1998a,b] or q-shift filters in Kingsbury [2000].We will focus here on the common-factor approach, developed in Selesnick [2001, 2002]. InSelesnick [2002] a numerical algorithm is proposed to compute the FIR filters associated to anapproximate Hilbert pair of orthogonal wavelet bases. Improvements of this method have beenproposed recently in Tay [2010], Murugesan and Tay [2014]. The approach of Selesnick [2001]is particularly attractive as it builds upon the usual orthogonal wavelet base constructionby solving a Bezout polynomial equation. Nevertheless, to the best of our knowledge, thevalidity of this specific construction have not been proved. Moreover the quality of theanalytic approximation have not been thoroughly assessed. The main goal of this paper isto fill these gaps. We also provide a short simulation study to numerically evaluate the qualityof analyticity approximation for specific common-factor wavelets.After recalling the definition of Hilbert pair wavelet filters, the construction of the Thiran’s common-factor wavelets following Thiran [1971], Selesnick [2002] is summarized in Section 2.Theoretical results are then developed to evaluate the impact of the Thiran’s common-factor degree L on the analytic property of the derived complex wavelet. In Section 3, an explicitformula to quantify the analytic approximation is derived. In addition, we provide a bounddemonstrating the improvement of the analytic property as L increases. These results apply toall wavelets obtained from FIR filters with Thiran’s common-factor . Of particular interest arethe orthogonal wavelet bases with perfect reconstruction. Section 4 is devoted to proving theexistence of such wavelets arising from filters with Thiran’s common-factor , which correspondto the wavelets introduced in Selesnick [2001, 2002]. Finally, in Section 5, some numericalexamples illustrate our findings. All proofs are given in the Appendices. Let ψ G and ψ H be two real-valued wavelet functions. Denote by (cid:98) ψ G and (cid:98) ψ H their Fouriertransform, (cid:98) ψ G ( ω ) = (cid:90) ψ G ( t ) e − i tω d ω . We say that ( ψ G , ψ H ) forms a Hilbert pair if (cid:98) ψ G ( ω ) = − i sign( ω ) (cid:98) ψ H ( ω ) , ω ) denotes the sign function taking values − , ω < ω = 0 and ω > ψ H ( t ) + i ψ G ( t ) is analytic since its Fouriertransform is only supported on the positive frequency semi-axis.Suppose now that the two above wavelets are obtained from the (real-valued) low-pass filters( g ( n )) n ∈ Z and ( h ( n )) n ∈ Z , using the usual multi-resolution scheme (see Daubechies [1992]).We denote their z-transforms by G ( · ) and H ( · ), respectively. In Selesnick [2001] andOzkaramanli and Yu [2003], it is established that a necessary and sufficient condition for( ψ G , ψ H ) to form a Hilbert pair is to satisfy, for all ω ∈ ( − π, π ), G (e i ω ) = H (e i ω )e − i ω/ . (1)Since e − i ω/ takes different values at ω = π and ω = − π , we see that this formula cannot holdif both G and H are continuous on the unit circle, which indicates that the construction ofHilbert pairs cannot be obtained with usual convolution filters and in particular with finiteimpulse response (FIR) filters. Hence a strict analytic property for the wavelet is not achievablefor a compactly supported wavelet, which is also a direct consequence of the Paley-Wienertheorem.However, for obvious practical reasons, the compact support property of the wavelet andthe corresponding FIR property of the filters must be preserved. Thus the strict analyticcondition (1) has to be relaxed into an approximation around the zero frequency, G (e i ω ) ∼ H (e i ω )e − i ω/ as ω → . (2)Several constructions have then been proposed to define approximate Hilbert pair wavelets,that is, pairs of wavelet functions satisfying the quasi analytic condition (2) [Tay, 2007].The common-factor procedure proposed in Selesnick [2002], is giving one solution to theconstruction of approximate Hilbert pair wavelets. This is the focus of the followingdevelopments. The common-factor procedure [Selesnick, 2002] is designed to provide approximate Hilbertpair wavelets driven by an integer L (cid:62) common factor transfer function F . Namely, the solution reads H ( z ) = F ( z ) D L ( z ) , (3) G ( z ) = F ( z ) D L (1 /z ) z − L , (4)where D L is the z transform of a causal FIR filter of length L , D L ( z ) = 1 + (cid:80) L(cid:96) =1 d ( (cid:96) ) z − (cid:96) ,such that e − i ωL D L (e − i ω ) D L (e i ω ) = e − i ω/ + O ( ω L +1 ) as ω → . (5)In Thiran [1971], a causal FIR filter satisfying this constraint is defined, the so-called maximallyflat solution given by (see also [Selesnick, 2002, Eq (2)]): d ( (cid:96) ) = ( − n (cid:18) L(cid:96) (cid:19) (cid:96) − (cid:89) k =0 / − L + k / k , (cid:96) = 1 , . . . , L. (6)3he cornerstone of our subsequent results is the following simple expression for D L ( z ), whichappears to be new, up to our best knowledge. Proposition 1.
Let L be a positive integer and D L ( z ) = 1 + (cid:80) Ln =1 d ( n ) z − n where thecoefficients ( d ( n )) n are defined by (6) . Then, for all z ∈ C ∗ , we have D L ( z ) = 12(2 L + 1) z − L (cid:104) (1 + z / ) L +1 + (1 − z / ) L +1 (cid:105) , (7) where z / denotes any of the two complex numbers whose squares are equal to z . Here C ∗ denotes the set of all non-zero complex numbers. Remark . In spite of the ambiguity in the definition of z / , the right-hand side in (7) isunambiguous because, when developing the two factors in the expression between squarebrackets, all the odd powers of z / cancel out. Remark . It is interesting to note that the closed form expression (7) of D L directly impliesthe approximation (5). Indeed, the right-hand side of (7) yields D L (e i ω ) = 12(2 L + 1) e − i ω ( L − / / (cid:2) (2 cos( ω/ L +1 + ( −
2i sin( ω/ L +1 (cid:3) = 2 L L + 1 e − i ω ( L − / / cos L +1 ( ω/
4) + O ( ω L +1 ) . It is then straightforward to obtain (5).
Proof.
See Section A.To summarize the common-factor approach, we use the following definition.
Definition ( Common-factor wavelet filters) . For any positive integer L and FIR filter withtransfer function F , a pair of wavelet filters { H , G } is called an L -approximate Hilbertwavelet filter pair with common factor F if it satisfies (3) and (4) with H (1) = G (1) = √ H (1) = G (1) = √ F (1) = √ D L (1) = √ L + 1)2 − L . (8)A remarkable feature in the choice of the common filter F is that it can be used to ensureadditional properties such as an arbitrary number of vanishing moments, perfect reconstructionor smoothness properties.First an arbitrary number M of vanishing moments is set by writing F ( z ) = Q ( z )(1 + 1 /z ) M , (9)with Q ( z ) the z -transform of a causal FIR filter (hence a real polynomial of z − ).4n additional condition required for the wavelet decomposition is perfect reconstruction. It isacquired when the filters satisfy the following conditions (see Vetterli [1986]): G ( z ) G (1 /z ) + G ( − z ) G ( − /z ) = 2 , (PR-G) H ( z ) H (1 /z ) + H ( − z ) H ( − /z ) = 2 . (PR-H)This condition is classically used for deriving wavelet bases ψ Gj,k = 2 j/ ψ G (2 j · − k ) and ψ Hj,k = 2 j/ ψ H (2 j · − k ), j, k ∈ Z , which are orthonormal bases of L ( R ). This will beinvestigated in Section 4. common-factor wavelets We now investigate the quasi-analyticity properties of the complex wavelet obtained fromHilbert pairs wavelet filters with the common-factor procedure.Let ( φ H ( · ) , ψ H ( · )) be respectively the father and the mother wavelets associated with the (low-pass) wavelet filter H . The transfer function H is normalized so that H (1) = √ (cid:98) φ H ( ω ) = ∞ (cid:89) j =1 (cid:104) − / H (e i2 − j ω ) (cid:105) , (10) (cid:98) ψ H ( ω ) = 2 − / H (e i ω/ ) (cid:98) φ H ( ω/ , (11)where H is the corresponding high-pass filter transfer function defined by H ( z ) = z − H ( − z − ) (see e.g. Selesnick [2001]). We also denote by ( φ G , ψ G ) the father and themother wavelets associated with the wavelet filter G . Equations similar to (10) and (11) holdfor (cid:98) φ G , and (cid:98) ψ G using G and G in place of H and H (see e.g. Selesnick [2001]).We first give an explicit expression of (cid:98) φ G and of (cid:98) ψ G with respect to (cid:98) φ H and (cid:98) ψ H . Theorem 2.
Let L be a positive integer. Let { H , G } be an L -approximate Hilbert waveletfilter pair. Let ( φ H , ψ H ) denote the father and mother wavelets defined by (10) and (11) anddenote ( φ G , ψ G ) the wavelets defined similarly from the filter G . Then, we have, for all ω ∈ R , (cid:98) φ G ( ω ) = e i β L ( ω ) (cid:98) φ H ( ω )e − i ω/ , (12) (cid:98) ψ G ( ω ) = i e i η L ( ω ) (cid:98) ψ H ( ω ) . (13) where α L ( ω ) = 2( − L arctan (cid:0) tan L +1 ( ω/ (cid:1) , (14) β L ( ω ) = ∞ (cid:88) j =1 α L (2 − j ω ) , (15) η L ( ω ) = − α L ( ω/ π ) + β L ( ω/ . (16) In (14) , we use the convention arctan( ±∞ ) = ± π/ so that α L is well defined on R . roof. See Section A.Following Theorem 2, we can write, for all ω ∈ R , (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) = (cid:16) − e i η L ( ω ) (cid:17) (cid:98) ψ H ( ω ) . (17)This formula shows that the quasi-analytic property and the Fourier localization of the complexwavelet ψ H + i ψ G can be respectively described by(a) how close the function 1 − e i η L is to the step function 2 R + (or − e i η L to the sign function);(b) how localized the (real) wavelet ψ H is in the Fourier domain.Property (b) is a well known feature of wavelets usually described by the behavior of thewavelet at frequency 0 (e.g. M vanishing moments implies a behavior in O ( | ω | M )) andby the polynomial decay at high frequencies. This behavior depends on the wavelet filter(see Villemoes [1992], Eirola [1992], Ojanen [2001]) and a numerical study of property (b) isprovided in Section 5.Note that, remarkably, property (a), only depends on L . Figure 1 displays the function1 − e i η L for various values of L . It illustrates the fact that as L grows, 1 − e i η L indeed getscloser and closer to the step function 2 R + . We can actually prove the following result whichbounds how close the Fourier transform of the wavelet ψ H + i ψ G is to 2 R + (cid:98) ψ H .Denote, for all ω ∈ R and A ⊂ R , the distance of ω to A by δ ( ω, A ) = inf {| ω − x | : x ∈ A } . (18) Theorem 3.
Under the same assumptions as Theorem 2, we have, for all ω ∈ R , (cid:12)(cid:12)(cid:12) (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) − R + ( ω ) (cid:98) ψ H ( ω ) (cid:12)(cid:12)(cid:12) = U L ( ω ) (cid:12)(cid:12)(cid:12) (cid:98) ψ H ( ω ) (cid:12)(cid:12)(cid:12) , where U L is a R → [0 , function satisfying, for all ω ∈ R , U L ( ω ) (cid:54) √ (cid:18) log (cid:18) max(4 π, | ω | )2 π (cid:19) + 2 (cid:19) (cid:18) − δ ( ω, π Z )max(4 π, | ω | ) (cid:19) L +1 . (19) Proof.
See Section A.This result provides a control over the difference between the Fourier transform (cid:98) ψ H + i (cid:98) ψ G of the complex wavelet and the Fourier transform 2 R + (cid:98) ψ H of the analytic signal associatedto ψ H . In particular, as L → ∞ , the relative difference U L = (cid:12)(cid:12)(cid:12) (cid:98) ψ H + i (cid:98) ψ G − R + (cid:98) ψ H (cid:12)(cid:12)(cid:12) / (cid:12)(cid:12)(cid:12) (cid:98) ψ H (cid:12)(cid:12)(cid:12) converges to zero exponentially fast on any compact subsets that do not intersect 4 π Z .6 . . . . . w - e i h L ( w ) L = 2L = 4L = 8L = 16 - p - p p p Figure 1: Plots of the function ω (cid:55)→ | − e i η L ( ω ) | for L = 2, 4, 8, 16. Let us now follow the path paved by Selesnick [2002] to select Q appearing in thefactorization (9) of the common factor F to impose M vanishing moments. First observethat, under (3), (4) and (9), the perfect reconstruction conditions (PR-G) and (PR-H) bothfollow from R ( z ) S ( z ) + R ( − z ) S ( − z ) = 2 , (20)where we have set R ( z ) = Q ( z ) Q (1 /z ) and S ( z ) = (2 + z + 1 /z ) M D L ( z ) D L (1 /z ).To achieve (20), the following procedure is proposed in Selesnick [2002], which follows theapproach in Daubechies [1992] adapted to the common factor constraint in (3). Step 1
Find R with finite, real and symmetric impulse response satisfying (20). Step 2
Find a real polynomial Q (1 /z ) satisfying the factorization R ( z ) = Q ( z ) Q (1 /z ).However, in Selesnick [2002], the existence of solutions R and Q is not proven, althoughnumerical procedures indicate that solutions can be exhibited. We shall now fill this gap andshow the existence of such solutions for any integers M, L (cid:62) R . Proposition 4.
Let L and M be two positive integers. Let D L be defined as in Proposition 1and let S ( z ) = (2 + z + 1 /z ) M D L ( z ) D L (1 /z ) . Then the two following assertions hold.(i) There exists a unique real polynomial r of degree at most M + L − such that R ( z ) = r (cid:16) z +1 /z (cid:17) satisfies (20) for all z ∈ C ∗ . ii) For any real polynomial p , the function R ( z ) = p (cid:16) z +1 /z (cid:17) satisfies (20) on z ∈ C ∗ ifand only if it satisfies p ( y ) = r ( y ) + s (1 − y ) q ( y ) , (21) where s ( y ) = y M L (cid:88) n =0 (cid:18) L + 12 n (cid:19) y n , (22) and q is any real polynomial satisfying q (1 − y ) = − q ( y ) .Proof. See Section B.Proposition 4 provides a justification of
Step 1 . In particular, a natural candidate for
Step 1 is R ( z ) = r (cid:16) z +1 /z (cid:17) . Now, by the Riesz Lemma (see e.g. [Daubechies, 1992, Lemma 6.1.3]),the factorization of Step 2 holds if and only if R ( z ) takes its values in R + on the unit circle { z ∈ C : | z | = 1 } , or equivalently r ( y ) (cid:62) y ∈ [0 , r ), checking this property theoreticallyfor all integers L, M is not yet achieved.Nevertheless we next prove that
Step 2 can always be carried out for any
L, M (cid:62)
1, at leastby modifying r into a polynomial p of the form (21) with a conveniently chosen q . Theorem 5.
Let L and M be two positive integers and let r and s be the polynomials definedas in Proposition 4. Then there exists a real polynomial q such that R ( z ) = [ r + s q ] (cid:16) z +1 /z (cid:17) is a solution of (20) and satisfies the factorization R ( z ) = Q ( z ) Q (1 /z ) where Q (1 /z ) , realpolynomial of z , does not vanish on the unit circle.Proof. See Section B.Proposition 4 and Theorem 5 allows one to carry out the usual program to the constructionof compactly supported orthonormal wavelet bases, as described in Daubechies [1992]. Hencewe get the following.
Corollary 6.
Let L and M be two positive integers. Let Q be as in Theorem 5. Define F asin (9) and let { H , G } be the L -approximate Hilbert wavelet filter pair associated to F . Thenthe wavelet bases ( ψ H,j,k ) and ( ψ G,j,k ) are orthonormal bases of L ( R ) . Observe that Theorem 5 states the existence of the polynomial Q but does not define it in aunique way. We explain why in the following remark. Remark . Since r in Proposition 4 is defined uniquely, it follows that, if we require that allthe roots of Q are inside the unit circle, there is at most one solution for Q with degree atmost K = M + L −
1, which correspond to the case q = 0. This solution, when it exists, isusually called the minimal phase, minimal degree solution. However we were not able to provethat r does not vanish on [0 , Q . Hence we instead prove the existence of solutions for Q byallowing q to be non-zero. 8 Numerical computation of approximate Hilbert waveletfilters
Let M and L be positive integers. Then, by Theorem 5, we can define the polynomial Q andderive from its coefficients the impulse response of the corresponding L -approximate Hilbertwavelet filter pair with M vanishing moments and perfect reconstruction.We now discuss the numerical computation of the coefficients of Q in the case where thepolynomial r defined by Proposition 4 does not vanish on [0 , r . Then the roots of r can also becomputed by a numerical solver and, as explained in Remark 3, if they do not belong on [0 , R ( z ) = r ((2 + z + 1 /z ) /
4) into Q ( z ) Q (1 /z ) by separating the roots conveniently.Taking all roots of modulus inferior to 1 leads to “mid-phase” wavelets. There are other waysof factorizing R , namely “min-phase” wavelets, see Selesnick [2002], leading to wavelets withFourier transform of the same magnitude but with different phases. This difference can beuseful in some multidimensional applications where the phase is essential.Hence the computation of the wavelet filters boils down to the numerical computation of thepolynomial r defined by Proposition 4. In Selesnick [2002], this computation is achieved byusing the following algorithm. • Let s = ( (cid:0) k M (cid:1) ) k =0 ,..., M and s = (cid:0) d L (0) . . . d L ( L ) (cid:1) (cid:63) (cid:0) d L ( L ) . . . d L (0) (cid:1) , where (cid:63) denotes the convolution for sequences. Then S ( z ) = (2 + z + 1 /z ) M D L ( z ) D L (1 /z ) = (cid:80) M + L ) n =0 s ( n ) z n − ( M + L ) with s = s (cid:63) s . The filter s has length 2( M + L ) + 1. • The filter r is such that s (cid:63) r is half-band. Let T denote the Toeplitz matrix associatedwith (cid:0) . . . s (cid:1) , vector of length 4( M + L ) + 1, that is, T k,j = s (1 + ( k − j )) if0 (cid:54) k − j (cid:54) M + L ) and T k,j = 0 else. We introduce C the matrix obtained bykeeping only the even rows of T , which has size (2( M + L ) − × (2( M + L ) − r is the solution of the equation Cr = b (23)with b = (cid:0) . . . . . . (cid:1) a 2( M + L ) − i.e. at ( M + L )-th position).We implemented this linear inversion method but it turned out that the corresponding linearequation is ill posed for too high values of M and L (for instance M = L = 7). For smallervalues of L and M , we recover the wavelet filters of the hilbert.filter program of the R -package waveslim computed only for ( M, L ) equal to (3,3), (3,5), (4,2) and (4,4), see Whitcher[2015]. 9 .2 A recursive approach to the computation of the Bezout minimal degreesolution
We propose now a new method for computing the L -approximate common-factor waveletpairs with M vanishing moments under the perfect reconstruction constraint. As explainedpreviously, this computation reduces to determining the coefficients of the polynomial r definedin Proposition 4. Our approach is intended as an alternative to the linear system resolutionstep of the approach proposed in Selesnick [2002]. Since our algorithm is recursive, to avoidany ambiguity, we add the subscripts L, M for denoting the polynomials r and s appearingin 4. That is, we set s L,M ( y ) = y M L (cid:88) n =0 (cid:18) L + 12 n (cid:19) y n and r L,M is the unique polynomial of degree at most M + L − B ( L, M )] r L,M (1 − y ) s L,M (1 − y ) + r L,M ( y ) s L,M ( y ) = (2 L + 1) − L − M +1 . We propose to compute r L,M for all L (cid:62) M (cid:62) Proposition 7.
Let L (cid:62) . Define y k,L = − tan (cid:18) π (2 k + 1)2(2 L + 1) (cid:19) , k ∈ { , · · · , L − } . (24) Then the solution r L, of the Bezout equation [ B ( L, is given by r L, ( y ) = (2 L + 1) − L +1 L − (cid:88) k =0 (cid:81) m (cid:54) = k ( y − (1 − y m,L )) s L, (1 − y k,L ) (cid:81) m (cid:54) = k ( y m,L − y k,L ) . (25) Moreover, for all M (cid:62) , we have the following relation between the solution of [ B ( L, M )] andthat of [ B ( L, M − : y r L,M ( y ) = r L,M − ( y ) − − L r L,M − (0) (1 − y ) s L,M − (1 − y ) . (26) Proof.
See Appendix C.This result provides a recursive way to compute r L,M by starting with r L, using theinterpolation formula (25) and then using the recursive formula (26) to compute r L, , r L, , . . . up to r L,M . In contrast to the method of Selesnick [2002] which consists in solving a (possiblyill posed) linear system, this method is only based on product and composition of polynomials.
We now provide some numerical results on the quality of the analyticity of the L -approximatedHilbert wavelet. All the numerical computations have been carried out by the method of10elesnick [2002] which seems to be the one used by practitioners (as in the software of Whitcher[2015]). Recall that as established in Theorem 3, for all ω ∈ R , (cid:12)(cid:12)(cid:12) (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) − R + ( ω ) (cid:98) ψ H ( ω ) (cid:12)(cid:12)(cid:12) = U L ( ω ) (cid:12)(cid:12)(cid:12) (cid:98) ψ H ( ω ) (cid:12)(cid:12)(cid:12) , where U L is displayed in Figure 2. Thus the quality of analyticity relies on the behavior of U L but also of (cid:98) ψ H ( ω ). First, (cid:98) ψ H ( ω ) goes to 0 when ω → M vanishingmoments given by (9). Secondly, | (cid:98) ψ H ( ω ) | decays to zero as | ω | goes to infinity. This last pointis verified numerically, by the estimation of the Sobolev exponents of ψ H using Ojanen [2001]’salgorithm. Values are given in Table 1. For M > R have the same exponents sincethe methods do not change the magnitude of (cid:98) ψ H + i (cid:98) ψ G .Table 1: Sobolev exponent estimated for ψ H functions. Dots correspond to configurationswhere numerical instability occurs in the numerical inversion of (23).M \ L 1 2 3 4 5 6 7 81 0.60 0.72 0.81 0.89 0.94 0.98 0.99 1.002 1.11 1.23 1.34 1.44 1.54 1.63 1.73 1.823 1.52 1.64 1.74 1.83 1.92 2.01 2.09 2.174 1.87 1.98 2.07 2.16 2.24 2.32 2.40 2.485 2.19 2.29 2.37 2.45 2.53 2.60 2.68 · · · · · · · · Figure 2 displays the overall shapes of the Fourier transforms (cid:98) ψ H of orthonormal waveletswith common-factor for various values of M and L . Their quasi-analytic counterparts (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) are plotted below in the same scales. It illustrates the satisfactory quality ofanalityc approximation. . . . M= 2 y H . . . w y H + i y G - p - p p p . . . M= 3 y H . . . w y H + i y G - p - p p p . . . M= 4 y H . . . w y H + i y G - p - p p p Figure 2: Top row: Plots of | (cid:98) ψ H | for M = 2(left), 3 (center), 4 (right) and L = 2 (black), 4(red), 8 (green). Bottom row: same for | (cid:98) ψ H + i (cid:98) ψ G | .11ay et al. [2006] propose two objective measures of quality based on the spectrum, E = max {| (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) | , ω < } max {| (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) | , ω > } and E = (cid:82) ω< | (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) | dω (cid:82) ω> | (cid:98) ψ H ( ω ) + i (cid:98) ψ G ( ω ) | dω . Numerical values of E and E are computed using numerical evaluations of (cid:98) ψ H on a grid,and, concerning E , using Riemann sum approximations of the integrals. Such numericalcomputations of E and E are displayed in Figure 3 for various values of M and L . Thefunctions E and E are decreasing with respect to L (which corresponds to the behaviour of U L ). They are also decreasing with respect to M (through the faster decay of (cid:98) ψ H around zeroand infinity). Moreover, the values illustrate the good analyticity quality of common-factor wavelets. For example, values appear to be lower than those of approximate analytic waveletsbased on Bernstein polynomials given in Tay et al. [2006]. . . . E L E + − − E L E M = 2M = 3M = 4M = 5M = 6
Figure 3: Plot of E and E with respect to L for different values of M . Approximate Hilbert pairs of wavelets are built using the common-factor approach. Specificfilters are obtained under perfect reconstruction conditions. They depend on two integerparameters L and M which correspond respectively to the order of the analytic approximationand the number of null moments. We demonstrate that the construction of such wavelets isvalid by proving their existence for any parameters L, M (cid:62)
1. Our main contribution in thispaper is to provide an exact formula of the relation between the Fourier transforms of the tworeal wavelets associated to the filters. This expression allows us to evaluate the analyticityapproximation of the wavelets, i.e. to control the presence of energy at the negative frequency.12his result may be useful for applications, where the approximated analytic properties of thewavelet have to be optimized, in addition to the usual localization in time and frequency.Numerical simulations show that these wavelets are easy to compute for not too large valuesof L and M , and confirm our theoretical findings, namely, that the analytic approximationquickly sharpens as L increases. This research did not receive any specific grant from funding agencies in the public, commercial,or not-for-profit sectors.
A Proofs of Section 3
A.1 Proof of Proposition 1
Proof of (7) . Notice that d ( L ) − z L D L ( z ) = (cid:80) Ln =0 d ( L − n ) d ( L ) z n and that for all n = 0 , . . . , L − d ( L − n ) d ( L ) = (cid:18) Ln (cid:19) L − (cid:89) (cid:96) = L − n (cid:96) + 32 L − (cid:96) − (cid:18) Ln (cid:19) (cid:32) n (cid:89) k =1 (2 k − (cid:33) − L (cid:89) (cid:96) = L − n +1 (2 (cid:96) + 1)= L ! n !( L − n )! 2 n n !(2 n )! (2 L + 1)!2 L L ! 2 L − n ( L − n )!(2 L − n + 1)!= (cid:18) L + 12 n (cid:19) It is then easy to check that d ( L ) − z L D L ( z ) = 12 (cid:16) (1 + z / ) L +1 + (1 − z / ) L +1 (cid:17) . The fact that d ( L ) = 1 / (2 L + 1) concludes the proof. A.2 Technical results on D We first establish the following result, which will be useful to handle ratios with D L (e iω ). Lemma 8.
Let L be a positive integer. Define D L as in Proposition 1. Then D L ( z ) does notvanish on the unit circle ( | z | = 1 ) and min z ∈ C : | z | =1 | D L ( z ) | = | D L ( − | = 2 L L + 1 < max z ∈ C : | z | =1 | D L ( z ) | = | D L (1) | = 2 L L + 1 . roof. Since D L (1 /z ) is a real polynomial of z , we have for all z ∈ C such that | z | = 1, | D L ( z ) | = D L ( z ) D L (1 /z ). Moreover, as shown in the proof of Proposition 4, if z = e θ with θ ∈ R , then | D L ( z ) | reads as in (43), which is minimal and maximal for cos( θ ) = 0 and 1,respectively.We now study z − L D L (1 /z ) D L ( z ) on the circle. Lemma 9.
For all z = e i ω with ω ∈ R , we have e − i ωL D L (e − i ω ) D L (e i ω ) = e − i ω/ α L ( ω ) , (27) where α L is the function defined on R by (14) .Proof. Observe that, for all z ∈ C ∗ , denoting by z / any of the two roots of z , z − L D L (1 /z ) D L ( z ) = z L (1 + z − / ) L +1 + (1 − z − / ) L +1 (1 + z / ) L +1 + (1 − z / ) L +1 = z − / (1 + z / ) L +1 + ( z / − L +1 (1 + z / ) L +1 − ( z / − L +1 Set now z = e i ω . We deduce thate − i ωL D L (e − i ω ) D L (e i ω ) = e − i ω/ e i ω (2 L +1) / cos( ω/ L +1 (1 + i( − L tan( ω/ L +1 )e i ω (2 L +1) / cos( ω/ L +1 (1 − i( − L tan( ω/ L +1 ) . The result then follows from the classical result a − i a = e arctan ( a ) with here a =( − L tan( ω/ L +1 . A.3 Proof of Theorem 2
Proof of equality (12) . Equation (10) provides the relation between (cid:98) φ H and H . The samerelation holds between (cid:98) φ G and G . It follows with Lemma 8, (3) and (4), that, for all ω ∈ R , (cid:98) φ G ( ω ) = (cid:98) φ H ( ω ) ∞ (cid:89) j =1 (cid:34) e − i ω − j L D L (e − i ω − j ) D L (e i ω − j ) (cid:35) . Applying Lemma 9, we get that, for all ω ∈ R , (cid:98) φ G ( ω ) = (cid:98) φ H ( ω ) ∞ (cid:89) j =1 e − i ω − j / α L ( ω − j ) = (cid:98) φ H ( ω ) exp − i ω/ ∞ (cid:88) j =1 − j + i ∞ (cid:88) j =1 α L ( ω − j ) . We thus obtain (12) using the definition of β L given by (15).14 roof of equality (13) . First observe that the relation between the high-pass filters G and H follows from that between the low-pass filter G and H , namely G ( z ) = ( − z ) L D L ( − z ) D L ( − /z ) H ( z ) . The relationship between (cid:98) ψ G and (cid:98) φ G is given by (11) (exchanging G and H ), yielding, for all ω ∈ R , (cid:98) ψ G ( ω ) = 2 − / ( − L e i ωL/ D L ( − e i ω/ ) D L ( − e − i ω/ ) H (e i ω/ ) (cid:98) φ G ( ω/ . We now replace (cid:98) φ G by the expression obtained in (12) and thanks to (11), (cid:98) ψ G ( ω ) = ( − L e i ωL/ D L ( − e i ω/ ) D L ( − e − i ω/ ) e − i ω/ e i β L ( ω/ (cid:98) ψ H ( ω ) . Since D L has a real impulse response and − i π = e − i π , Lemma 9 gives that, for all ω ∈ R ,( − L e i ωL/ D L ( − e i ω/ ) D L ( − e − i ω/ ) = e − i L ( ω/ π ) D L (e − i( ω/ π ) ) D L (e i( ω/ π ) ) = i e i ω/ − i α L ( ω/ π ) . Hence, we finally get that, for all ω ∈ R , (cid:98) ψ G ( ω ) = i e − i α L ( ω/ π )+i β L ( ω/ (cid:98) ψ H ( ω ) . (13) is proved. A.4 Proof of Theorem 3
Approximation of − e i η L We first state a simple result on the function e i α L . Lemma 10.
Let L be a positive integer. The function α L defined by (14) is (4 π ) -periodic.Moreover e i α L is continous on R and we have, for all ω ∈ R , (cid:12)(cid:12)(cid:12) e i α L ( ω ) − I ( ω ) (cid:12)(cid:12)(cid:12) (cid:54) √ L +1 ( ω ) , (28) where I ( ω ) = (cid:40) if ω ∈ [ − π, π ) + 4 π Z − otherwise, (29) and ∆( ω ) := min (cid:0) | tan( ω/ | , | tan( ω/ | − (cid:1) (30) Proof.
By definition (14), α L is (4 π )-periodic and continuous on R \ (2 π + 4 π Z ). Moreover, atany of its discontinuity points in 2 π + 4 π Z , α L jumps have height 2 π . Hence e i α L is continousover R . 15e now prove (28). We will in fact show the following more precise bounds, valid for all ω ∈ R . | cos( α L ( ω )) − | (cid:54) | tan( ω/ | L +1) , (31) | cos( α L ( ω )) + 1 | (cid:54) | tan( ω/ | − L +1) , (32) | sin( α L ( ω )) | (cid:54) L +1 ( ω ) . (33)The bounds (31) and (32) easily follow from the identitycos( α L ( ω )) = cos(2arctan(tan( ω/ L +1 )) = 1 − tan L +1) ( ω/ L +1) ( ω/ . The bound (33) follows from the identitysin( α L ( ω )) = sin(2arctan(tan L +1 ( ω/ L +1 ( ω/ L +1) ( ω/ . The proof is concluded.Observe that by (15) and the definition of η L in (16), e i η L can be expressed directly from e i α L ,namely as e i η L ( ω ) = e − i α L ( ω/ π ) (cid:89) j (cid:62) e i α L ( − j − ω ) . A quite natural question is to determine the function 1 − e i η L obtained when e i α L is replacedby its large L approximation I . This is done in the following result. Lemma 11.
Define the (4 π ) -periodic rectangular function I by (29) . Then, for all ω ∈ R \{ } ,we have − I ( ω/ π ) (cid:89) j (cid:62) I (cid:0) − j − ω (cid:1) = 2 R + ( ω ) . (34) Proof.
Note that the function ω (cid:55)→ I ( ω + π ) is the right-continuous, (4 π )-periodic functionthat coincides with the sign of ω on ω ∈ [ − π, π ) \ { } . It is then easy to verify that, bydefinition of I , we have, for all ω ∈ R , I ( ω ) = I (2 ω + π ) I ( ω + π )= I (2 ω + π ) I ( ω + π ) (35)= −I (2 ω + π ) −I ( ω + π ) . (36)(By periodicity of I , it only suffices to check the first equality on ω ∈ [ − π, π ), the two otherequalities follow, since I takes values in {− , } .) Now, from the previous assertion, we have,for all ω <
0, that I ( ω − j + π ) = 1 for large enough j , and thus (35) implies (cid:89) j (cid:62) I (2 − j ω ) = (cid:89) j (cid:62) I (2 − ( j − ω + π ) I (2 − j ω + π ) = I ( ω + π ) , ω >
0, since −I ( ω − j + π ) = 1 for large enough j , (36) implies (cid:89) j (cid:62) I (2 − j ω ) = (cid:89) j (cid:62) −I (2 − ( j − ω + π ) −I (2 − j ω + π ) = −I ( ω + π ) . Identity (34) follows.We can now derive the main result of this section.
Proposition 12.
Let L be a positive integer. The function η L defined by (14) , (15) and (16) satisfies the following bound, for all ω ∈ R , (cid:12)(cid:12)(cid:12) − e i η L ( ω ) − R + ( ω ) (cid:12)(cid:12)(cid:12) (cid:54) √ (cid:32) ∆ L +1 ( ω/ π ) + ∞ (cid:88) k =1 ∆ L +1 (2 − k − ω ) (cid:33) , (37) where ∆ is defined by (30) .Proof. We have, for all ω ∈ R and J (cid:62) J (cid:89) j =1 e i α L (2 − j ω ) − J (cid:89) j =1 I (2 − j ω ) = J (cid:88) k =1 a k,J ( ω ) , where we denote a k,J ( ω ) = k − (cid:89) j =1 e i α L (2 − j ω ) · (cid:16) e i α L (2 − k ω ) − I (2 − k ω ) (cid:17) · J (cid:89) j = k +1 I (2 − j ω ) , with the convention (cid:81) ( . . . ) = (cid:81) JJ +1 ( . . . ) = 1. Since α L is real valued and I is valued in {− , } , it follows that, for all ω ∈ R and J (cid:62) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J (cid:89) j =1 e i α L (2 − j ω ) − J (cid:89) j =1 I (2 − j ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) J (cid:88) k =1 | a k,J ( ω ) | (cid:54) J (cid:88) k =1 (cid:12)(cid:12)(cid:12) e i α L (2 − k ω ) − I (2 − k ω ) (cid:12)(cid:12)(cid:12) . Applying Lemma 10 yields for all ω ∈ R and J (cid:62) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J (cid:89) j =1 e i α L (2 − j ω ) − J (cid:89) j =1 I (2 − j ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) √ (cid:32) J (cid:88) k =1 ∆ L +1 (2 − k ω ) (cid:33) . Letting J → ∞ and applying the definition of β L , we deduce that, for all ω ∈ R , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e i β L ( ω ) − ∞ (cid:89) j =1 I (2 − j ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) √ ∞ (cid:88) k =1 ∆ L +1 (2 − k ω ) . (38)By definition of η L , since α L and β L are real valued and I is valued in {− , } , we have, forall ω ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e i η L (2 ω ) − I ( ω + π ) ∞ (cid:89) j =1 I (2 − j ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) (cid:12)(cid:12)(cid:12) e i α L ( ω + π ) − I ( ω + π ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e i β L ( ω ) − ∞ (cid:89) j =1 I (2 − j ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ω ∈ R , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e i η L (2 ω ) − I ( ω + π ) ∞ (cid:89) j =1 I (2 − j ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:54) √ (cid:32) ∆ L +1 ( ω + π ) + ∞ (cid:88) k =1 ∆ L +1 (2 − k ω ) (cid:33) . The bound (37) then follows from Lemma 11.
Study of the upper bound
The objective is to simplify the right-hand side of (37) to obtain the form given in (19). Thefollowing lemma essentially gives some interesting properties of the function ∆.
Lemma 13.
Let ∆ be the function defined by (30) . Then ∆ is an even (2 π ) -periodic function,increasing and bijective from [0 , π ] to [0 , . It follows that, for all ω ∈ R , ∆( ω ) = tan (cid:18) π − π δ ( ω, π + 2 π Z )) (cid:19) (cid:54) − π δ ( ω, π + 2 π Z ) , (39) where δ is the function defined in equation (18) .Proof. The proof is straightforward and thus omitted.Note that the upper bound in (39) decreases from 1 to 0 as δ ( ω, π + 2 π Z ) increases from 0 to π . Since ∆ takes its values in [0 ,
1) on R \ ( π + 2 π Z ), Lemma 10 shows that, out of the set π + 2 π Z , e i α L uniformly converges to the (4 π )-periodic rectangular function I as L → ∞ .We will use the following bound. Lemma 14.
For all ω ∈ ( − π/ , π/ and L (cid:62) , we have ∞ (cid:88) j =0 | tan | L +1 (2 − j ω ) (cid:54) | tan | L +1 ( ω ) . Proof.
It suffices to prove the inequality for ω ∈ (0 , π/ x (cid:55)→ x − tan( x ) is increasing on [0 , π/
4) and so is x (cid:55)→ x − tan L +1 ( x ) for L (cid:62)
0. Hence wehave, for all ω ∈ (0 , π/ ∞ (cid:88) j =0 tan L +1 (2 − j ω ) = ∞ (cid:88) j =0 − j ω (cid:0) − j ω (cid:1) − tan L +1 (2 − j ω ) (cid:54) ∞ (cid:88) j =0 − j ω ( ω ) − tan L +1 ( ω )= 2 tan L +1 ( ω ) . The proof is concluded. 18e also have the following lemma.
Lemma 15.
For all ω ∈ R , we have δ ( ω/ π, π + 2 π Z ) (cid:62) − δ ( ω, π Z ) , (40) δ (2 − j ω, π + 2 π Z ) (cid:62) − j δ ( ω, π Z ) for all integer j (cid:62) . (41) Proof.
The bound (40) is obvious. To show (41), take x ∈ π + 2 π Z . Then, for all ω ∈ R and j (cid:62)
2, we have (cid:12)(cid:12) − j ω − x (cid:12)(cid:12) = 2 − j (cid:12)(cid:12) ω − j x (cid:12)(cid:12) and, since 2 j x ∈ π Z , we get (41).We are now able to give a more concise upper bound. Lemma 16.
Let ∆ be defined by (30) . Then, for all ω ∈ R , we have ∞ (cid:88) j =1 ∆ L +1 (2 − j − ω ) (cid:54) (cid:18) log (cid:18) max(4 π, | ω | )2 π (cid:19) + 1 (cid:19) (cid:18) − δ ( ω, π Z )max(4 π, | ω | ) (cid:19) L +1 . (42) Proof.
Denote ι ( ω ) = min (cid:8) j (cid:62) | ω | − j − < π (cid:9) (cid:54) log (cid:18) max (4 π, | ω | )2 π (cid:19) . ∞ (cid:88) j =1 ∆ L +1 (2 − j − ω ) (cid:54) ι ( ω ) − (cid:88) j =1 ∆ L +1 (2 − j − ω ) + (cid:88) j (cid:62) ι ( ω ) | tan | L +1 (2 − j − ω )Lemma 14 gives that, for all ω ∈ R , (cid:88) j (cid:62) ι ( ω ) | tan | L +1 (2 − j − ω ) (cid:54) | tan | L +1 (2 − ι ( ω ) − ω ) = 2∆ L +1 (2 − ι ( ω ) − ω ) . The last two bounds yield, for all ω ∈ R , ∞ (cid:88) j =1 ∆ L +1 (2 − j − ω ) (cid:54) ( ι ( ω ) + 1) (cid:32) sup (cid:54) j (cid:54) ι ( ω ) ∆(2 − j − ω ) (cid:33) L +1 . Note that Lemma 13 and (41) implysup (cid:54) j (cid:54) ι ( ω ) ∆(2 − j − ω ) (cid:54) − π − ι ( ω ) − δ ( ω, π Z ) . The above bound on ι ( ω ) then gives (42).We can now conclude with the proof of the main result. Proof of Theorem 3.
By Lemma 13 and (40), we have, for all ω ∈ R ,∆( ω/ π ) (cid:54) − π δ ( ω, π Z ) (cid:54) − δ ( ω, π Z )max(4 π, | ω | ) . Using this bound, (17), Proposition 12 and Lemma 16, we get (19).19
Proofs of Section 4
The following lemma will be useful.
Lemma 17.
Let L be a positive integer. The complex roots of the polynomial (cid:101) s ( x ) = (cid:80) Ln =0 (cid:0) L +12 n (cid:1) x n belong to R − .Proof. Observe that for all z ∈ C , (cid:101) s ( z ) = ((1 + z ) L +1 + (1 − z ) L +1 ). Thus if (cid:101) s ( z ) = 0with z = x + i y and ( x, y ) ∈ R , we necessarily have that | z | = (1 + x ) + y is equal to | − z | = (1 − x ) + y , and thus x = 0 and z ∈ R − . Proof of Proposition 4.
By Proposition 1, we have, for all θ ∈ R , D L (e θ ) D L (e − θ ) = 14(2 L + 1) (cid:12)(cid:12)(cid:12) (1 + e i θ ) L +1 + (1 − e i θ ) L +1 (cid:12)(cid:12)(cid:12) = 2 L +1) L + 1) (cid:104) cos L +1) ( θ/
2) + sin L +1) ( θ/ (cid:105) = 2 L (2 L + 1) (cid:20) (1 + cos( θ )) L +1 + (1 − cos( θ )) L +1 (cid:21) = 2 L (2 L + 1) L (cid:88) n =0 (cid:18) L + 12 n (cid:19) cos n ( θ ) . (43)Note that if z = e θ with θ ∈ R , then 2 + z + 1 /z = 2(1 + cos(2 θ )) = (2 cos( θ )) . By definitionof S , we obtain that, for all θ ∈ R , S (e θ ) = 2 M cos M ( θ ) × L (2 L + 1) L (cid:88) n =0 (cid:18) L + 12 n (cid:19) cos n ( θ )= 2 M +2 L (2 L + 1) s (cos ( θ )) , where s is the polynomial defined by (22). Looking for a solution R of (20) in the form R ( z ) = U (cid:16) z +1 /z (cid:17) with U real polynomial and focusing on z = e θ with θ ∈ R , we obtainthe equation U (1 − y ) s (1 − y ) + U ( y ) s ( y ) = C ( L, M ) . (44)where we have denoted y = sin ( θ ) = 1 − (2 + z + 1 /z ) / ∈ [0 ,
1] and C ( L, M ) =(2 L + 1)2 − M − L +1 / . Reciprocally, any such polynomial U provides a solution R ( z ) = U (cid:16) z +1 /z (cid:17) of (20) for z = e θ with θ ∈ R and then for all z ∈ C ∗ by analytic extension.Since the complex roots of s are valued in the set R − of non-positive real numbers (seeLemma 17), we get that s (1 − y ) and s ( y ) are prime polynomials of degree L + M . Thus theBezout Theorem allows us to describe the couples of real polynomials ( U, V ) solutions of theequation V ( y ) s (1 − y ) + U ( y ) s ( y ) = C ( L, M ) . U is a solution of (44) if and only if ( U, V ) is a solution of the Bezout equation with V ( y ) = U (1 − y ). Now, by uniqueness of the solution of the Bezout equation such that both U and V have degrees at most L + M −
1, we see that this solution must satisfy V ( y ) = U (1 − y )(since otherwise ( V (1 − y ) , U (1 − y )) would provide a different solution). Hence we obtain aunique solution U = r of (44) of degree at most L + M −
1, which proves Assertion (i).Other solutions (
U, V ) of the Bezout equation are obtained by taking U ( y ) = r ( y )+ s (1 − y ) q ( y )with q any polynomial. Looking for such a solution of (44), we easily get that it is one if andonly if q satisfies q (1 − y ) = − q ( y ). The proof of Assertion (ii) is concluded. Proof of Theorem 5.
By Proposition 4, the given R is a solution of (20) provided that q satisfies q (1 − y ) = − q ( y ), which we assume in the following. As explained above, the factorizationholds if and only if R is non-negative on the unit circle, or, equivalently, if r ( y ) + s (1 − y ) q ( y )is non-negative for y in (0 , q around 1 /
2, this is equivalent to have,for all y ∈ [1 / , r ( y ) + s (1 − y ) q ( y ) and r (1 − y ) − s ( y ) q ( y ) (cid:62) . Since s ( y ) > y ∈ (0 ,
1) and using that (44) holds with U = r , we finally obtain thatthe claimed factorization holds if and only if, for all y ∈ [1 / , − r ( y ) s (1 − y ) (cid:54) q ( y ) (cid:54) − r ( y ) s (1 − y ) + C ( L, M ) s ( y ) s (1 − y ) . (45)We first note that for all y ∈ [1 / , / ( s ( y ) s (1 − y )) (cid:62) / ( s (1) s (1 / >
0. Hence the upperbound condition in (45) is away from the lower bound by at least a positive constant over y ∈ [0 , / U = r at y = 1 / r (1 / s (1 /
2) = C ( L, M ) . It follows that, for y = 1 /
2, (45) reads − C ( L, M ) s (1 / (cid:54) q (1 / (cid:54) C ( L, M ) s (1 / , This is compatible with q (1 /
2) = 0 inherited by the antisymmetric property of q around1 /
2. We conclude by applying the Stone-Weierstrass theorem to obtain the existence of a realpolynomial q satisfying (45) for all y ∈ [1 / , q ( y ) = − q (1 − y ) for all y ∈ R . C Proofs of Section 5
We start with a result more precise than Lemma 17.
Lemma 18.
Let L be a positive integer. The complex roots of the polynomial (cid:101) s ( y ) = (cid:80) Ln =0 (cid:0) L +12 n (cid:1) y n are the y ,L , . . . , y L − ,L defined in (24). roof. Recall that that (cid:101) s ( z ) = (cid:2) (1 + z ) L +1 + (1 − z ) L +1 (cid:3) and that (cid:101) s ( z ) = 0 is equivalentto z (cid:54) = 1 and (cid:16) z − z (cid:17) L +1 = −
1, that is1 + z − z = e i( π +2 kπ ) / (2 L +1) , k ∈ {− L, · · · , L } . There is no such z for k = L and for k ∈ {− L, · · · , L − } , this is the same as z = e i( π +2 kπ ) / (2 L +1) −
11 + e i( π +2 kπ ) / (2 L +1) = i tan (cid:18) π (2 k + 1)2(2 L + 1) (cid:19) . Taking the square and keeping only k = 0 , . . . , L to get distinct roots we get the result. Proof of Proposition 7.
Using Lemma 18 and the Bezout equation [ B ( L, k ∈ { , · · · , L − } r L, (1 − y k,L ) = (2 L + 1) − L +1 s L, (1 − y k,L ) . Since the polynomial r L, has degree at most L − L distinct points1 − y k,L , we deduce its explicit expression given in Proposition 7 by a standard interpolationformula.We conclude with the proof of the recursive relation (26). Since for any M (cid:62) s L,M ( y ) = ys L,M − , the polynomial r L,M satisfies r L,M (1 − y ) × [(1 − y ) s L,M − (1 − y )] + r L,M ( y ) × [ ys L,M − ( y )] = (2 L + 1) − M − L +1 . (46)We deduce that 2 yr L,M ( y ) satisfies equation [ B ( L, M − yr L,M ( y ) = r L,M − ( y ) + s L,M − (1 − y ) q ( y ) , (47)where q is a polynomial satisfying q (1 − y ) = − q ( y ). Since the degree of r L,M is at most L + M − s L,M − is L + M −
1, we get that q takes the form q ( y ) = q (0)(1 − y ) and it onlyremains to determine q (0). Note that s L,M (1) = 2 L , so (47) yields q (0) = − − L r L,M − (0)and we finally obtain (26). References
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